This article addresses the global synchronisation problem of a network of coupled Lur'e systems from the perspective of global synchronised region. A decomposition approach is proposed to convert the synchronisation of high-dimensional Lur'e networks into the test of a set of matrix inequalities whose dimensions are the same as a single Lur'e node. The notion of global synchronised region is then introduced and analysed. A necessary and sufficient condition is derived for the existence of the inner-linking matrix to guarantee a desirable unbounded synchronised region. A multi-step design procedure is given for constructing such an inner-linking matrix, which maintains a favourable decoupling property. Furthermore, the global H-infinity synchronised region is characterised for evaluating the performance of a Lur'e network subject to external disturbances. The effectiveness of the theoretical results is demonstrated through a network of Chua's circuits.